Vector or Cross Product

The vector product between two vectors is a vector quantity represented as x or a and is defined by the following equation  

Finally, the vector product can also be written as a matrix determinant  

The scalar products involving the three unit vectors are, therefore, given by 

These vectors, thus, constitute an orthonormal set with respect to scalar multiplication. To express the scalar product A B in terms of the components of A and B, write 

Consider a point P rotating counterclockwise about a vertical axis through the origin, as shown in Fig. 11.3. Let r be the vector from the origin to point P, and let v be the instantaneous linear velocity of the point as it moves around a circle of radius r sin0. A rotation is conventionally represented by an axial 

FIGURE 11.3 Rotation of point P about vertical axis, represented by the vector product relation v = X r. 

FIGURE 11.4 Right-hand rules. Left direction of axial vector representing counterclockwise rotation. Right direction of vector product A X B. 

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Conversely, the vector (or cross) product of the same two vectors (vixv ) is defined as a vector V3 in the direction perpendicular to the plane of Vi and V2, whose magnitude is equal to the product of the absolute values of the two vectors and the sine of the angle a between them, or... 

The vector or cross product of two unit vectors is defined by ... 

The vector or cross product of two unit vectors was defined by (C.94), hence in cylindrical coordinates the following relations are valid ... 

The inner and outer products of two vectors produce a scalar and a matrix, respectively. Two 3-vectors may also be multiplied together to generate a new 3-vector, a procedure called a vector or cross product, with the result given by eq. (16.15). 

The vector or cross product determines a so-called axial vector with orthogonal orientation from two spatial vectors. The vector product is anti-commutative ... 

There are at least two ways to multiply vectors. The iirst way is to find the scalar or dot product between the two vectors. The second way is to find the vector, or cross product, between the two vectors. We shall consider scalar multiplication first. 

Vector Multiplication. The vector or cross product is defined by the equation... 

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